Digital Signal Processing Reference
In-Depth Information
Given a real periodic signal
x(t),
a
harmonic series
for this signal is defined as
q
C
k
e
jkv
0
t
;
*
x(t) =
C
k
= C
-k
.
(4.11)
a
k=-
q
The frequency is called the
fundamental frequency
or the
first harmonic,
and
the frequency is called the
k
th
harmonic
. If the coefficients and the signal
in (4.11) are related by an equation to be developed later, this harmonic se-
ries is a
Fourier series
. For this case, the summation (4.11) is called the
complex
exponential form,
or simply the exponential form, of the Fourier series; the coef-
ficients are called the
Fourier coefficients
. Equation (4.10) is an example of a
Fourier series in the exponential form. We next derive a second form of the
Fourier series.
The general coefficient
v
0
kv
0
C
k
x(t)
C
k
C
k
in (4.11) is complex, as indicated in Table 4.1, with
C
-k
equal to the conjugate of
C
k
.
The coefficient
C
k
can be expressed as
C
k
= ƒC
k
ƒ e
ju
k
,
-
q
6
k 6
q
.
C
-k
= C
*
,
with
Since
it follows that
u
-k
=-u
k
.
For a given value of
k
, the sum of the two terms of the same frequency
kv
0
in (4.11) yields
C
-k
e
-jkv
0
t
+ C
k
e
jkv
0
t
= ƒC
k
ƒ e
-ju
k
e
-jkv
0
t
+ ƒC
k
ƒ e
ju
k
e
jkv
0
t
= ƒC
k
ƒ [e
-j(kv
0
t +u
k
)
+ e
j(kv
0
t +u
k
)
]
= 2 ƒC
k
ƒ cos(kv
0
t + u
k
).
(4.12)
Hence, given the Fourier coefficients
C
k
,
we can easily find the
combined trigono-
metric form
of the Fourier series:
q
x(t) = C
0
+
a
2 ƒC
k
ƒ cos(kv
0
t + u
k
).
(4.13)
k= 1
A third form of the Fourier series can be derived from (4.13). From Appendix A,
we have the trigonometric identity
cos(a + b) = cos
a cos
b - sin a sin b.
(4.14)
The use of this identity with (4.13) yields
q
2
ƒ
C
k
ƒ
cos(kv
0
t + u
k
)
x(t) = C
0
+
a
k= 1
q
[2
ƒ
C
k
ƒ
cos
u
k
cos
kv
0
t - 2
ƒ
C
k
ƒ
sin u
k
sin kv
0
t].
= C
0
+
a
(4.15)
k= 1
